4.5.1 Tensor Argument Slot Position

Tensor Argument Slot Position defines where arguments are placed in tensor operations, crucial for understanding tensor algebra structure and function application.

Tensor Argument Slot Position is the ordinal placement of a given slot within a tensor's argument slot structure, that is, the numerical index (first, second, third, and so on) identifying exactly where in the ordered sequence of inputs a particular slot sits. Slot position is what distinguishes otherwise identical slots that share the same vector space and variance, and it is the coordinate that all order-sensitive tensor operations, such as permutation, symmetrization, and contraction, must reference explicitly.

Formal Definition

Position as an Index into the Slot Sequence

For a tensor represented as a multilinear map of arity $k$ with an ordered argument slot set

S = (s_{1}, s_{2}, \dots, s_{k})

the position of a slot $s_m$ is the index $m \in {1, \ldots, k}$. Position is entirely independent of the vector space a slot draws from and independent of its variance; two slots at different positions can be bound to the same space with the same variance and still be logically distinct because of where they sit in the sequence.

Positional Substitution

When arguments $v_1, \ldots, v_k$ are supplied to $T$, the value $v_m$ must be substituted into the slot at position $m$:

T (v_{1}, \dots, v_{m}, \dots, v_{k})

This positional binding is what makes multilinearity a slot-by-slot statement: the linearity condition for slot $m$ only concerns variation of the $m$-th argument, holding the arguments at all other positions fixed.

Position and Order Sensitivity

Why Position Matters

Because a general multilinear map is not required to be symmetric, the value $T(u, v)$ can differ from $T(v, u)$ even when $u$ and $v$ come from the same space. The distinction between these two evaluations is entirely a matter of slot position: the same pair of vectors, assigned to swapped positions, can produce different outputs. Position is therefore not a bookkeeping convenience but a substantive part of the tensor's definition.

Position Under Permutation

A permutation $\sigma$ of ${1, \ldots, k}$ acts on slot positions by relabeling them, producing a new map

T_{σ} (v_{1}, \dots, v_{k}) = T (v_{σ (1)}, \dots, v_{σ (k)})

A tensor is symmetric across a set of positions if $T_\sigma = T$ for every permutation $\sigma$ restricted to those positions, and antisymmetric if $T_\sigma = \operatorname{sgn}(\sigma), T$. Both notions are stated purely in terms of what happens to the output when slot positions are exchanged.

Position in Index Notation and Operations

Positional Correspondence With Index Slots

Once a basis is fixed, each slot position corresponds to exactly one index in the component array of the tensor. A change in which position holds an upper versus lower index constitutes a genuinely different tensor type ordering, even for the same values of $r$ and $s$, because conventions for which physical index a formula refers to (e.g. the first upper index versus the second) depend on position.

Position-Specific Contraction

Contracting a tensor requires specifying exactly which position holds the contravariant slot and which holds the covariant slot being paired; the contraction operator is therefore always described with explicit reference to slot positions, such as "contract the first upper index with the second lower index," since the identity of the operation depends on more than just the type $(r,s)$ alone.

Position Preservation Under Tensor Product

When two tensors are combined via the tensor product, the resulting map's slot positions are formed by concatenating the original two ordered slot sequences, with the first tensor's positions coming before the second tensor's positions. This convention is what allows unambiguous recovery of each original tensor's slots inside the combined higher-arity object.

Summary of Key Points

Slot position is the ordinal index of a slot within the ordered argument sequence of a multilinear map.
Two slots can share the same vector space and variance yet be distinct solely due to differing position.
Symmetry and antisymmetry are defined as invariance (up to sign) of the output under permutation of slot positions.
Once a basis is fixed, slot position corresponds directly to the position of an index in the tensor's component array.
Operations such as contraction and tensor product require explicit reference to slot positions to be unambiguous.